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This paper presents a series of results on the interplay between quantumestimation, cloning and finite de Finetti theorems. First, we consider themeasure-and-prepare channel that uses optimal estimation to convert M copiesinto k approximate copies of an unknown pure state and we show that thischannel is equal to a random loss of all but s particles followed by cloningfrom s to k copies. When the number k of output copies is large with respect tothe number M of input copies the measure-and-prepare channel converges indiamond norm to the optimal universal cloning. In the opposite case, when M islarge compared to k, the estimation becomes almost perfect and themeasure-and-prepare channel converges in diamond norm to the partial trace overall but k systems. This result is then used to derive de Finetti-type resultsfor quantum states and for symmetric broadcast channels, that is, channels thatdistribute quantum information to many receivers in a permutationally invariantfashion. Applications of the finite de Finetti theorem for symmetric broadcastchannels include the derivation of diamond-norm bounds on the asymptoticconvergence of quantum cloning to state estimation and the derivation of boundson the amount of quantum information that can be jointly decoded by a group ofk receivers at the output of a symmetric broadcast channel.

On quantum estimation, quantum cloning and finite quantum de Finetti theorems